#483125

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10,14,19,12,21,28,18,15,20,31,36,29,17,13,30)(11,22,25,34,27,35,40,38,26,37,39,16,23,33,24,32);(2,5)(3,7)(9,11)(10,16)(12,23)(13,25)(14,24)(15,26)(17,32)(18,35)(19,37)(20,34)(21,38)(22,36)(27,29)(28,39)(30,33)(31,40);(2,5)(6,8)(9,12,15,29)(10,17,20,21)(11,23,26,27)(13,14,28,31)(16,32,34,38)(18,19,30,36)(22,35,37,33)(24,39,40,25);(9,12,15,29)(10,17,20,21)(11,23,26,27)(13,14,28,31)(16,32,34,38)(18,19,30,36)(22,35,37,33)(24,39,40,25)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309650, #72309660, #72309670, …
maximal normal subgroup of	#72309650, #72309660, #72309670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390321);